Answer to Question #264596 in Real Analysis for Nikhil

Question

The set { x∈ R : x≤ 5} is not a closed set in R.

True or false

Answer 1

Let us show that the set $O=\{ x∈ \R : x> 5\}$ is open set in $\R$. Let $a\in O$ be arbitrary, and let $\varepsilon_a=a-5>0.$ Then the open ball $B_{\varepsilon_a}(a)=(a-\varepsilon_a,a+\varepsilon_a)$ is contained in $O.$ It follows that each point $a\in O$ is inner, and hence the set $O$ is open. Therefore, the set $\{ x∈ \R : x≤ 5\}=\R\setminus O$ is a closed set in $\R$.

Answer: false